Chapter XX Quantum theory of light

Chapter XX Quantum theory of light

At the end of the 19-th century, physics was at its most confidence
situation. Classical phyics, as formulated in Newton’s law of mechanics
and Maxwell’s theory of electromagnetism, have proved very successful
in solving every problem.
→At that time there seemed to be no question for which physics could
not provide an answer !!!
But then it came as a great shock when some simple phenomena were
observed which could not be explained by classical physics
→a new theory, quantum theory, was developed at the beginning of the
20-th century...

 At the end of the 19-th century, physics was at its most confidence
situation. Classical phyics, as formulated in Newton’s law of mechanics
and Maxwell’s theory of electromagnetism, have proved very successful
in solving every problem.
→ At that time there seemed to be no question for which physics could
not provide an answer !!!
 But then it came as a great shock when some simple phenomena were
observed which could not be explained by classical physics
→ a new theory, quantum theory, was developed at the beginning of the
20-th century
 We begin our study of quantum physics by two following phenomena:
• Blackbody radiation
• Photoelectric effect
We will see what were the failures of classical physics and how a new
theory had been developed.

§1. Blackbody radiation. Planck’s theory of radiation:
• Heat bodies emit electromagnetic radiation in the infra-red region of the
spectrum (see the next slide). In this region the radiation is not visible.
• As the temperature of a body is increased to any value, the body begins
to glow red and then white, emitting visible electromagnetic radiation.
(an example is the variation of the radiation of a filament of electric lamp
when the electric current varies).
• Observation of the spectrum emitted by a solid shows that the radiation
extends over a continous range of frequency. Such a spectrum is called
a continuum.
1.1 Experimental laws of blackbody radiation:
1.1.1 Stephan-Boltzmann law:
• It was observed that the intensity of the radiation emitted from a
body increases rapidly with increasing temperature of the body.

• The relation between intensity and temperature is given by the following
formula:
I =  T4 ,

where I → the average power of radiation per unit surface area,
→ a fundamental physics constant called the Stephan-Boltzmann
constant = 5,67 x 10-8 W m-2 K-4
 a dimensionless number (0 <  1) called the emissivity, which
→ <
depends on the nature of the radiating surface.
It is found that for any surface the absorption is the exact
reverse of the emission process. It means that the ability of
emission is proportional to that of absorption:
  0, the surface has low emissivity and low ability of absorption
→

 → 1, the surface has high emissivity and high ability of absorption
The idealized case, when  1, such a body is called absolute blackbody.
=
A blackbody absorbs completely all the incident radiation and gives no
reflecting radiation (that is why the body is black !).

• Therefore we have for a blackbody: I = T4
This is the Stephan-Boltzmann law for a blackbody
• Remark: For a blackbody the total intensity
depends only on absolute temperature
• A cavity with a small aperture is an example
of a blackbody. Electromagnetic radiation
entering the cavity is eventually absorbed
after successive reflections → the cavity is
a perfect absorber of e-m radiation.
1.1.2 Wien displacement law:
Note that the intensity I in the Stephan-Boltzmann is the total intensity,
that is, the radiation intensity for all wavelengths.
Denote by I( dthe intensity corresponding to wavelengths in the
)
interval  and  d  can write
+ we 
I d
I( )
0

I( is the distribution function
)
over all wavelengths. It is called
the spectral emittance.
• The form of I( depends on
)
temperature. By experimental
measures one has IT ( shown
)
in the picture.
• Each curve has a peak at 
= m
It is observed that as the
temperature T increases, the
peak grown larger, and shifts to
shorter wavelengths.
• By experiment Wien showed
that  is inversely proportional
m
to T, and
 T = 2.90 x 10-3 m.K
m This is Wien displacement law

1.2 Rayleigh–Jeans formula . Ultraviolet catastrophe:
• Rayleigh and Jeans attemped to explain the observed blackbody spectrum
on the base of the concept that radiation is a e-m wave. They derived the
following formula for the intensity distribution:
2ckT
I ( 
)
 4
• Comparing the Rayleigh-Jeans
spectrum with experiment one
can see that:
The R-J formula agrees well
with experiment at large 
IT()
But there is a serious
disagreement at small 
The unrealistic behavior of
the Rayleigh-Jeans distribution
at short wavelengths is known in
physics as “ultraviolet catastrophe”.

A more impressive indication of the complete failure of the Rayleigh-
Jeans spectrum is the result on the total intensity:
 
d  2ckT 1 
I   d 2  4 
I( ) ckT lim  3 
0 0
 
3  0  
It means that I →  , an unceptable result !!!
The blackbody radiation spectrum could not be explained
by classical physics
1.2 Planck’s theory of radiation:
The discrepancy between experiment and theory was resolved in 1900
by Planck, by introducing a postulate which was revolutionary with
respect to certain concepts of classical physics.

1.2.1 Planck’s postulate and radiation law:
Planck’s postulate: “Electromagnetic radiation consists of simple harmonic
oscillations which can possess only energies
 nh (n = 0, 1, 2, 3, ….)
=
where n is the frequancy of the oscillation, and h is an universal constant”
Energy level diagrams for classical sipmple harmonic
oscillations and for that obeying Planck’s postulates.
The constant h is called the Planck constant. It’s value was determined
by fitting theoretical consequences with experiment data (see below).

Equations (*), (**) and
fits the Planck law curve
with experiment leads to
the same result for the value
of the Planck’s constant:
h = 6.626x10-34 J.s
 Therefore, the experimental
laws for blackbody radiation
IT()
could have a satisfactory
explanation with the Planck’s
theory. The root idea is that
electromagnetic energy
emitted by bodies can have
only discrete values n.h The comparison of Planck’s spectrum
→ one say that energy is
with experiment at T = 1646 0K
quantized. The discrete nature
of energy is the foundation of quantum theory.
One more remark: The large wavelength limit of the Planck’s spectrum
formula is the Rayleigh’s formula (recall e x ≈1 + x with small x). This means
that at very long wavelengths (very small quanta energies), quantum effects
become unimportant.

§2.Photoelectric effect. Einstein’s theory of light:
2.1 Photoelectric effect:
 Consider a metal surface. Electrons in
a metal are “bound” by the binding
Binding
potential energy. Introduce the potential
quantity called “work function”.
,
 is the minimum amount of energy an

induvidual electron has to gain to escape:
• E <  the electron is confined inside metal
→ (E: the energy
• E >  the electron can escape from metal.
→ of the electron ).
 If you shine light on the metal surface, electrons absorb energy from
the incident light, and have enough energy to escape. Eshtablish on the
metal surface a electric field → the escaped electrons create an current
that is called photoelectric current
How will the photoelectric current depend on the intensity I and
the frequency  the incident light ?

of

2.2 Einstein’s quantum theory of light:
To overcome the difficulty of classical physics, Albert Einstein introduced
the quantum theory of light, and developed the correct analysis of the
photoelectric effect in 1905.
2.2.1 Einstein’s postulates:
By analogy to the earlier Planck’s theory of radiation, the postulates of
Einstein’s theory is as follows:
 Light consists of small “packages” of energy called photons or
light quanta.
 The energy  a photon is  h 
of =  where h is the Plank constant.
In vacuum  h hc/
= =
2.2.2 Analysis of the photoeffect by the quantum theory:
A photon arriving at the surface is absorbed by an electron (one by one).
After that the electron gets all the photon’s energy (h)

For an electron we can write the following equation:
mv2
h   max

2
The energy of the electron The energy part The remaining part, the kinetic
after absoption of photon for escaping from energy of the motion outside metal
the metal surface
Kinetic energy of the electron outside metal is telled by the value Vstop
in the experiment:
mv 2
K max  max eV stop eVstop  
h 
2
By this analysis it is understandable that
• Energy of electrons emitted depends on frequency, not intensity
• The limit frequency  for the photoeffect is determined by the
0
equation h = 
0 
 Under this light frequency the electron has not
enough energy to escape.

§3. Compton scattering:
Compton scattering (Compton effect) is a phenomenon that provides
additional direct confimation of the quantum nature of light, and
particularly, of X-rays.
3.1 Experimental results:
• The wavelength of the
incoming monochromatic
X-rays: 
• The wavelength of the
scattering X-rays:  ’
Experimental observations
discovered a shift in wavelength:  ≠ (A.H. Compton 1923).
’
The wavelength shift of scattered X-rays is called the Compton effect.
• It is found that the wavelength difference   -  varies with
 
’
the scattering angle according to the equation